
theorem Th49:

:: 1.11. THEOREM, (1) => (2b), p. 147
  for T being Lawson complete continuous TopLattice
  for S being full non empty SubRelStr of T
  st ex X being Subset of T st X = the carrier of S & X is closed
  holds S is directed-sups-inheriting
proof
  let T be Lawson complete continuous TopLattice;
  let S be full non empty SubRelStr of T;
  given X being Subset of T such that
A1: X = the carrier of S and
A2: X is closed;
  let Y be directed Subset of S;
  assume Y <> {};
  then reconsider D = Y as directed non empty Subset of T by YELLOW_2:7;
  set N = Net-Str D;
  assume ex_sup_of Y,T;
  the mapping of N = id Y by Th32;
  then rng the mapping of N = Y;
  then Lim N c= Cl X by A1,Th27,WAYBEL19:26;
  then
A3: Lim N c= X by A2,PRE_TOPC:22;
  sup D in Lim N by Th35;
  hence thesis by A1,A3;
end;
